Modelos_Estatisticos-2024-08-10
Helena R. S. D’Espindula
2024-08-10
Regressão Linear
### Modelos estatísticos - Aula de modelos lineares
### Pacotes usados na aula.
require("ISLR")## Carregando pacotes exigidos: ISLRrequire("ggplot2")## Carregando pacotes exigidos: ggplot2require("GGally")## Carregando pacotes exigidos: GGally## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2require("leaps") ## seleção de variaveis## Carregando pacotes exigidos: leapsrequire("car")## Carregando pacotes exigidos: car## Carregando pacotes exigidos: carDataoptions(device = X11)Recomendações de leitura
[…]
Objetivo
Num modelo de regressão linear, a relação entre uma variável aleatória de interesse (variável resposta) e um conjunto de preditores (variáveis explicativas) é definida por uma função linear envolvendo os preditores e um conjunto de parâmetros associados (coeficientes ou $$′s ou parametros).
“Composição”
- Variaveis Resposta
- Variaveis Explicativas
Exemplo:
- Aplicação 1: Preço de Imóveis
- Variável Resposta: Preço do imóvel
- Variáveis Explicativas: Tamanho do imóvel (m2), número de quartos, localização (bairro), idade do imóvel, presença de garagem
- Aplicação 2: Desempenho Escolar
- Variável Resposta: Nota final em uma disciplina
- Variáveis Explicativas: Número de horas estudadas por semana, frequência nas aulas, nota em provas anteriores, participação em atividades extraclasse, apoio familar
- Aplicação 3: Vendas de um produto
- Variável Resposta: Vendas mensais de um produto
- Variáveis Explicativas: Preço do produto, gastos em marketing, número de vendedores, promoções realizadas, condições econômicas.
- Aplicação 4: Consumo de Energia
- Variável Resposta: Consumo de energia elétrica (kWh)
- Variáveis Explicativas: Tamanho da residência (m2), número de moradores, uso de eletrodomésticos, temperatura média na região, tipo de aquecimento
- Aplicação 5: Rendimento de Investimentos
- Variável Resposta: Rendimento de uma carteira de investimentos
- Variáveis Explicativas: Taxa de juros, porcentagem de ações vs. renda fixa, valor inicial investido, taxa de inflação, duração do investimento
Modelo de regressão linear
Seja \(y\) a variável resposta e \(x1, x2, ..., xk\) os \(k\) preditores. O modelo de regressão linear fica definido por: \[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_k x_k + \epsilon \]
onde:
- \(\beta_0\) é o intercepto do modelo (se todos os \(x\)s e \(\epsilon\) forem 0 $ y = _0$ );
- \(\beta_1, \beta_2, ... \beta_k\) são parâmetroa do modelo, que correspondem aos efeitos lineares de cada um dos \(k\) preditores;
- \(\epsilon\) representa os erros aleatórios, específicos para cada indivíduo. É a parte de \(y\) que não é explicada pela regressão linear.
Erros
Embora sejam não observáveis e não explicados pelo modelo, na regressão linear assumimos as seguintes propriedades para os erros:
- Os erros têm média zero;
- Os erros têm variância constante;
- Os erros para dois indivíduos quaisquer são não correlacionados;
- Os erros têm distribuição normal.
Este conjunto de suposições é usualmente denotado por \(\epsilon \sim N (0, \sigma^2)\).
Regressão Linear Simples
Como caso particular da regressão linear temos a regressão linear simples, que se caracteriza por considerar um único preditor:
\[ y = \beta_0 + \beta_1 x + \epsilon, \space \space \space \space \space \epsilon \sim N (0, \sigma^2) \]
Regressão Linear Polinomial
Outro modelo de regressão linear bastante usual é o modelo de regressão polinomial, que permite explicar uma relação não linear entre a resposta e o(s) preditor(es):
\[
y = \beta_0 + \beta_1 x + \beta_2 x^2 + ... + \beta_k x^k + \epsilon,
\space \space \space \space \space
\epsilon \sim N (0, \sigma^2)
\] 
Fig 02
Fig 03
Fig 04
Fig 05
Fig 06
Um breve histórico
- Século 19: Desenvolvimento da teoria de mínimos quadrados, que serve de base para o ajuste de modelos de regressão.
- A teoria de mínimos quadrados teve origem na Física motivada, dentre outros, por problemas na área navegação (século 18).
- Ao longo do século 19, modelos lineares e o método de mínimos quadrados passaram a ser utilizados em outras ciências, baseados em modelos pré-estabelecidos, ou apenas em evidências empíricas.
- O termo regressão foi introduzido por Francis Galton em 1875, baseado no princípio da “regressão à média”.
Regressão linear
- Os dois objetivos principais de uma análise de regressão são os
seguintes:
- Exploratório- Analisar a relação funcional entre uma variável aleatória (resposta) e um conjunto de variáveis preditoras.
- Preditivo- Predizer valores não observados da variável resposta para valores especificados das variáveis preditoras.
Exemplo- Consumo de combustível
- Neste exemplo vamos analisar os dados da base Auto, disponíveis na biblioteca ISLR do R.
- A base de dados dispõe de informações técnicas de 392 modelos de automóveis das décadas de 1970 e 1980, como consumo de combustível. potência do motor, dimensões dentre outras.
- O objetivo aqui é ajustar modelos de regressão linear que expliquem o consumo de combustível com base em características do modelo.
- Neste primeiro momento, para fins ilustrativos vamos considerar apenas duas variáveis de cada vez, embora o mais usual seja considerar múltiplas variáveis conjuntamente numa análise de regressão.
summary(Auto)## mpg cylinders displacement horsepower weight
## Min. : 9.00 Min. :3.000 Min. : 68.0 Min. : 46.0 Min. :1613
## 1st Qu.:17.00 1st Qu.:4.000 1st Qu.:105.0 1st Qu.: 75.0 1st Qu.:2225
## Median :22.75 Median :4.000 Median :151.0 Median : 93.5 Median :2804
## Mean :23.45 Mean :5.472 Mean :194.4 Mean :104.5 Mean :2978
## 3rd Qu.:29.00 3rd Qu.:8.000 3rd Qu.:275.8 3rd Qu.:126.0 3rd Qu.:3615
## Max. :46.60 Max. :8.000 Max. :455.0 Max. :230.0 Max. :5140
##
## acceleration year origin name
## Min. : 8.00 Min. :70.00 Min. :1.000 amc matador : 5
## 1st Qu.:13.78 1st Qu.:73.00 1st Qu.:1.000 ford pinto : 5
## Median :15.50 Median :76.00 Median :1.000 toyota corolla : 5
## Mean :15.54 Mean :75.98 Mean :1.577 amc gremlin : 4
## 3rd Qu.:17.02 3rd Qu.:79.00 3rd Qu.:2.000 amc hornet : 4
## Max. :24.80 Max. :82.00 Max. :3.000 chevrolet chevette: 4
## (Other) :365ggplot(Auto, aes(x = year)) + geom_histogram()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot(Auto, aes(x = year, y = mpg)) + geom_point() +
stat_smooth(method = "lm") +
theme_bw(base_size = 14)## `geom_smooth()` using formula = 'y ~ x'
plot1 <- ggplot(Auto, aes(x = horsepower, y = mpg)) + geom_point() + theme_bw(base_size = 14)
plot1
plot1 + coord_trans(x="log2", y="log2")
Fig 07
Inicialmente, vamos ajustar um modelo de regressão linear que permita explicar o consumo de combustível (mpg- variável resposta) em função do ano de lançamento do modelo (yearvariável explicativa).
A Figura 7 sugere relação linear crescente entre o consumo de combustível e o ano de lançamento do modelo.
O modelo de regressão linear para esse par de variáveis fica especificado por: \[ mpg = \beta_0 + \beta_1 \times year + \epsilon, \] onde \(\beta_0\) e \(\beta_1\) são os parâmetros do modelo (intercepto e inclinação da reta de regressão) e \(\epsilon\) representa os erros aleatórios.
O ajuste da regressão linear consiste na estimação dos parâmetros do modelo (β0 e β1), com base nos dados amostrais, que produzem a reta de regressão que melhor se ajusta aos dados.
O método usual de estimação dos parâmetros de uma regressão linear é o método de mínimos quadrados, que será estudado adiante.
Aplicando o método de mínimos quadrados, obtemos os parâmetros estimados \(\hat{\beta_0} = −70.01\) e \(\hat{\beta_1} = 1.23\), produzindo a seguinte reta de regressão ajustada: \[ \hat{mpg} = −70.01 + 1.23 \times year \]
Fig 08
Fig 09
[continua….]
Exemplo prático- Taxa de formados
- Na parte prática desta aula vamos analisar dados sobre a taxa de formados em \(n=777\) universidades norte-americanas no ano de 1995.
- O objetivo é ajustar um modelo de regressão linear que permita explicar a taxa de alunos formados em função das características das universidades e de seus alunos.
- Uma breve descrição das variáveis explicativas é apresentada na sequência.
Variaveis
Grad.Rate: Taxa de alunos formados (resposta);Apps: Número de alunos inscritos;Accept: Número de alunos aceitos;Enroll: Número de novos alunos matriculados;Top10perc: Percentual de novos estudantes entre os 10% melhores no ensino médio;Top25perc: Percentual de novos estudantes entre os 25% melhores no ensino médio;F.Undergrad: Número de alunos em período integral nos cursos de graduação;P.Undergrad: Número de alunos em período parcial nos cursos de graduação;Outstate: Número de alunos bolsistas de outros estados;Room.Board: Gastos de hospedagem e alimentação;Books: Gastos com materiais bibliográficos;Personal: Gastos com recursos humanos;PhD: Percentual de professores com doutorado;Terminal: Percentual de professores com grau terminal;S.F.Ratio: Razão alunos/professor;perc.alumni: Percentual de ex-alunos que contribuem com donativos;Expend: Gasto educacional por aluno.- Privada ou publica
CODIGO
### Carregamento e visualização inicial da base
data("College") ### Carregando a base
#help("College") ### Acessando a documentação
head(College,10) ### Visualizando as dez primeiras linhas## Private Apps Accept Enroll Top10perc Top25perc
## Abilene Christian University Yes 1660 1232 721 23 52
## Adelphi University Yes 2186 1924 512 16 29
## Adrian College Yes 1428 1097 336 22 50
## Agnes Scott College Yes 417 349 137 60 89
## Alaska Pacific University Yes 193 146 55 16 44
## Albertson College Yes 587 479 158 38 62
## Albertus Magnus College Yes 353 340 103 17 45
## Albion College Yes 1899 1720 489 37 68
## Albright College Yes 1038 839 227 30 63
## Alderson-Broaddus College Yes 582 498 172 21 44
## F.Undergrad P.Undergrad Outstate Room.Board Books
## Abilene Christian University 2885 537 7440 3300 450
## Adelphi University 2683 1227 12280 6450 750
## Adrian College 1036 99 11250 3750 400
## Agnes Scott College 510 63 12960 5450 450
## Alaska Pacific University 249 869 7560 4120 800
## Albertson College 678 41 13500 3335 500
## Albertus Magnus College 416 230 13290 5720 500
## Albion College 1594 32 13868 4826 450
## Albright College 973 306 15595 4400 300
## Alderson-Broaddus College 799 78 10468 3380 660
## Personal PhD Terminal S.F.Ratio perc.alumni Expend
## Abilene Christian University 2200 70 78 18.1 12 7041
## Adelphi University 1500 29 30 12.2 16 10527
## Adrian College 1165 53 66 12.9 30 8735
## Agnes Scott College 875 92 97 7.7 37 19016
## Alaska Pacific University 1500 76 72 11.9 2 10922
## Albertson College 675 67 73 9.4 11 9727
## Albertus Magnus College 1500 90 93 11.5 26 8861
## Albion College 850 89 100 13.7 37 11487
## Albright College 500 79 84 11.3 23 11644
## Alderson-Broaddus College 1800 40 41 11.5 15 8991
## Grad.Rate
## Abilene Christian University 60
## Adelphi University 56
## Adrian College 54
## Agnes Scott College 59
## Alaska Pacific University 15
## Albertson College 55
## Albertus Magnus College 63
## Albion College 73
## Albright College 80
## Alderson-Broaddus College 52dim(College) ### Acessando a dimensão da base## [1] 777 18summary(College) ### Resumo das variáveis## Private Apps Accept Enroll Top10perc
## No :212 Min. : 81 Min. : 72 Min. : 35 Min. : 1.00
## Yes:565 1st Qu.: 776 1st Qu.: 604 1st Qu.: 242 1st Qu.:15.00
## Median : 1558 Median : 1110 Median : 434 Median :23.00
## Mean : 3002 Mean : 2019 Mean : 780 Mean :27.56
## 3rd Qu.: 3624 3rd Qu.: 2424 3rd Qu.: 902 3rd Qu.:35.00
## Max. :48094 Max. :26330 Max. :6392 Max. :96.00
## Top25perc F.Undergrad P.Undergrad Outstate
## Min. : 9.0 Min. : 139 Min. : 1.0 Min. : 2340
## 1st Qu.: 41.0 1st Qu.: 992 1st Qu.: 95.0 1st Qu.: 7320
## Median : 54.0 Median : 1707 Median : 353.0 Median : 9990
## Mean : 55.8 Mean : 3700 Mean : 855.3 Mean :10441
## 3rd Qu.: 69.0 3rd Qu.: 4005 3rd Qu.: 967.0 3rd Qu.:12925
## Max. :100.0 Max. :31643 Max. :21836.0 Max. :21700
## Room.Board Books Personal PhD
## Min. :1780 Min. : 96.0 Min. : 250 Min. : 8.00
## 1st Qu.:3597 1st Qu.: 470.0 1st Qu.: 850 1st Qu.: 62.00
## Median :4200 Median : 500.0 Median :1200 Median : 75.00
## Mean :4358 Mean : 549.4 Mean :1341 Mean : 72.66
## 3rd Qu.:5050 3rd Qu.: 600.0 3rd Qu.:1700 3rd Qu.: 85.00
## Max. :8124 Max. :2340.0 Max. :6800 Max. :103.00
## Terminal S.F.Ratio perc.alumni Expend
## Min. : 24.0 Min. : 2.50 Min. : 0.00 Min. : 3186
## 1st Qu.: 71.0 1st Qu.:11.50 1st Qu.:13.00 1st Qu.: 6751
## Median : 82.0 Median :13.60 Median :21.00 Median : 8377
## Mean : 79.7 Mean :14.09 Mean :22.74 Mean : 9660
## 3rd Qu.: 92.0 3rd Qu.:16.50 3rd Qu.:31.00 3rd Qu.:10830
## Max. :100.0 Max. :39.80 Max. :64.00 Max. :56233
## Grad.Rate
## Min. : 10.00
## 1st Qu.: 53.00
## Median : 65.00
## Mean : 65.46
## 3rd Qu.: 78.00
## Max. :118.00### Vamos considerar Grad.Rate (taxa de formados) como a variável resposta na nossa análise. Começamos a análise com alguns gráficos.
ggplot(College, aes(x = Grad.Rate)) + geom_histogram() +
theme_bw(base_size = 14)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
### Distribuição das taxas de formados
ggplot(College, aes(x = Top10perc, y = Grad.Rate)) + geom_point() +
geom_smooth(method = "loess") +
theme_bw(base_size = 14)## `geom_smooth()` using formula = 'y ~ x'
### Taxas de formados versus percentual de alunos entre os 10% melhores
### no ensino médio.
ggplot(College, aes(x = Outstate, y = Grad.Rate)) + geom_point() +
geom_smooth(method = "loess") +
theme_bw(base_size = 14)## `geom_smooth()` using formula = 'y ~ x'
### Taxas de formados versus investimentos externos.
ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
geom_smooth(method = "loess") +
theme_bw(base_size = 14)## `geom_smooth()` using formula = 'y ~ x'
### Taxas de formados versus porcentagens de ex-alunos contribuintes.ggpairs(College, proportions = "auto") ### Matriz de gráficos de dispersão. - DEMORA PARA RODAR
#para salvar o ultimo grafico
ggsave("plot_grande.png", device = "png", width = 100, height = 80, units = "cm")
# tamanho maximo 50 in ou 126 cm
Grafico pesado
ggcorr(College[,-1], label = TRUE, label_round = 2) ### Correlograma.
ggsave("plot_colorido.png", device = "png", width = 30, height = 20, units = "cm")
Grafico colorido
### Parte 1 - Ajuste dos modelos lineares. Comecemos com o caso de apenas uma
### variável explicativa (no caso, perc.alumni)
### Para ajustar modelos lineares no R usamos a função lm. Vamos consultar
### a documentação da função.
#help('lm')Multipla (1 var resposta) e Multivarida (mais var respostas) são coisas diferentes
### Ajuste da regressão linear simples (assumindo relação linear entre a taxa de formados e o percentual de ex-alunos contribuintes)
ajuste1 <- lm(Grad.Rate ~ perc.alumni, data = College) ###(resposta ~ var.explicativas)
ajuste1##
## Call:
## lm(formula = Grad.Rate ~ perc.alumni, data = College)
##
## Coefficients:
## (Intercept) perc.alumni
## 49.9863 0.6805\[ \hat{Grad.Rate} = 49,9863 + 0,6805 \times perc.alumni \] chapeu indica estimado?
summary(ajuste1)##
## Call:
## lm(formula = Grad.Rate ~ perc.alumni, data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -58.247 -9.513 0.043 9.362 54.404
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 49.98633 1.12345 44.49 <2e-16 ***
## perc.alumni 0.68049 0.04338 15.69 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.98 on 775 degrees of freedom
## Multiple R-squared: 0.241, Adjusted R-squared: 0.24
## F-statistic: 246.1 on 1 and 775 DF, p-value: < 2.2e-16### O percentual de ex-alunos contribuintes tem efeito positivo, e
### estatisticamente significativo na taxa de formados.### Vamos visualizar o ajuste do modelo
ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
stat_smooth(method = "lm") +
theme_bw(base_size = 14)## `geom_smooth()` using formula = 'y ~ x'
### Vamos investigar possível efeito quadrático do percentual de contribuíntes na taxa de formados. Para isso, adicionamos ao preditor o termo quadrático da variável explicativa, da seguinte forma:
ajuste2 <- lm(Grad.Rate ~ perc.alumni + I(perc.alumni^2), data = College)
summary(ajuste2)##
## Call:
## lm(formula = Grad.Rate ~ perc.alumni + I(perc.alumni^2), data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -57.444 -9.146 0.042 9.448 53.446
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 45.896244 1.877789 24.442 < 2e-16 ***
## perc.alumni 1.085948 0.155614 6.978 6.42e-12 ***
## I(perc.alumni^2) -0.007652 0.002821 -2.712 0.00683 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.91 on 774 degrees of freedom
## Multiple R-squared: 0.2481, Adjusted R-squared: 0.2462
## F-statistic: 127.7 on 2 and 774 DF, p-value: < 2.2e-16### O termo quadrático é estatisticamente significativo, indicando que a
### relação entre as variáveis não é linear. Vamos dar um passo além, e
### incluir o termo de terceira ordem para o percentual de contribuintes
### (modelo cúbico).\[ \hat{Grad.Rate} = 45.896244 + 1.085948 \times perc.alumni -0.007652 \times perc.alumni^2 \]
de acordo com os valores de p é significativo, então manteria essa variavel
ajuste22 <- lm(formula = Grad.Rate ~ ., data = College)
summary(ajuste22)##
## Call:
## lm(formula = Grad.Rate ~ ., data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -53.897 -7.132 -0.292 7.213 54.056
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.8736716 4.8480858 6.987 6.15e-12 ***
## PrivateYes 3.3813758 1.6965147 1.993 0.046605 *
## Apps 0.0012984 0.0004418 2.939 0.003390 **
## Accept -0.0006961 0.0008627 -0.807 0.419995
## Enroll 0.0021593 0.0023081 0.936 0.349814
## Top10perc 0.0548964 0.0717587 0.765 0.444501
## Top25perc 0.1351288 0.0549667 2.458 0.014179 *
## F.Undergrad -0.0004712 0.0004008 -1.176 0.240138
## P.Undergrad -0.0014836 0.0003902 -3.802 0.000155 ***
## Outstate 0.0010174 0.0002334 4.359 1.49e-05 ***
## Room.Board 0.0019143 0.0005908 3.240 0.001246 **
## Books -0.0022205 0.0029168 -0.761 0.446739
## Personal -0.0016635 0.0007698 -2.161 0.031000 *
## PhD 0.0872827 0.0568102 1.536 0.124859
## Terminal -0.0747023 0.0623172 -1.199 0.231002
## S.F.Ratio 0.0758222 0.1593102 0.476 0.634254
## perc.alumni 0.2793343 0.0491750 5.680 1.91e-08 ***
## Expend -0.0004565 0.0001542 -2.961 0.003163 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.75 on 759 degrees of freedom
## Multiple R-squared: 0.4615, Adjusted R-squared: 0.4495
## F-statistic: 38.27 on 17 and 759 DF, p-value: < 2.2e-16## PrivateYes -> 1 se privada (YES); 0 se publica(No)
## Nesse modelo (ajustado com todas as outras variaveis) as escolas privadas tem p<0.05, portanto ser privada tem um impacto no (maior) número de formados (3.3 pontos percentuais)
ajuste23 <- lm(formula = Grad.Rate ~ Private , data = College)
summary(ajuste23)##
## Call:
## lm(formula = Grad.Rate ~ Private, data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -53.998 -10.042 0.002 11.002 49.002
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 56.042 1.112 50.406 <2e-16 ***
## PrivateYes 12.956 1.304 9.937 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 16.19 on 775 degrees of freedom
## Multiple R-squared: 0.113, Adjusted R-squared: 0.1119
## F-statistic: 98.74 on 1 and 775 DF, p-value: < 2.2e-16## Num modelo só com a privada ela (por "acaso") tambem teve p<0.05 mas com um impacto diferente (12.9 pontos percentuais)confint(ajuste22) ### Intervalo de confiaça## 2.5 % 97.5 %
## (Intercept) 24.3564214869 43.3909217410
## PrivateYes 0.0509572771 6.7117943696
## Apps 0.0004312157 0.0021656167
## Accept -0.0023897796 0.0009975317
## Enroll -0.0023717122 0.0066902689
## Top10perc -0.0859726333 0.1957655121
## Top25perc 0.0272239131 0.2430337318
## F.Undergrad -0.0012581212 0.0003156791
## P.Undergrad -0.0022496029 -0.0007176822
## Outstate 0.0005592115 0.0014756792
## Room.Board 0.0007545476 0.0030741345
## Books -0.0079464883 0.0035055569
## Personal -0.0031746541 -0.0001524155
## PhD -0.0242410359 0.1988064637
## Terminal -0.1970367830 0.0476322483
## S.F.Ratio -0.2369186844 0.3885630948
## perc.alumni 0.1827990643 0.3758695198
## Expend -0.0007590979 -0.0001538281Intervalo de confiança para a resposta média e para uma predição
Considere interesse em estimar a resposta média em um ponto \(x′_0 = (1, x_{01], x_{02}, ..., x_{0k})\), ou seja, \(E (y |x_0)\).
A estimativa pontual é dada pelo valor ajustado pelo modelo em \(x_0\): \[ \widehat{E(y|x_0)} = \hat{y_0} = x′_0 \hat{\beta} \] O estimador apresentado é não viciado para a real resposta média, com variância:
[CONTINUA…]
\[ \hat{Grad.Rate} = 33,87 + 3,38 \times (Private = Yes) + 0.0012984 \times (Apps) \]
Variavel Dummie: (relevel)
Sexo: F, M -> Sexo Feminino (1 se feminino, 0 se masculino)
Escolaridade: Sem_Escolaridade, EF, EM, ES ->
- EF (1 ou 0)
- EM (1 ou 0)
- ES (1 ou 0)
- Sendo 1 TEM e 0 Não tem, consequentemente os 3 zeros é Sem_Escolaridade
\[ \hat{y} = \hat{\beta_0} + \hat{\beta_1} \times EF + \hat{\beta_2} \times EM + \hat{\beta_3} \times ES \] - Sem escolaridade: \(\hat{y} = \hat{\beta_0}\)
Ens. Fund: \(\hat{y} = \hat{\beta_0} + \hat{\beta_1}\)
Ens. Med: \(\hat{y} = \hat{\beta_0} + \hat{\beta_2}\)
Ens. Sup: \(\hat{y} = \hat{\beta_0} + \hat{\beta_3}\)
Se for EM/EF: \[ \hat{y} = (\hat{\beta_0} + \hat{\beta_1}) - (\hat{\beta_0} + \hat{\beta_2}) = \hat{\beta_1} - \hat{\beta_2} \]
\[ y = \beta_0 + \beta_1 x + \varepsilon \] \[ \varepsilon = (y - (\beta_0 + \beta_1 x)) \] \[ residuo = y - (\hat{\beta_0} + \hat{ \beta_1 } x) \] Erro é o “real”, residuo é o estimado.
ajuste3 <- lm(Grad.Rate ~ perc.alumni + I(perc.alumni^2) + I(perc.alumni^3), data = College)
summary(ajuste3)##
## Call:
## lm(formula = Grad.Rate ~ perc.alumni + I(perc.alumni^2) + I(perc.alumni^3),
## data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -56.571 -9.123 0.005 9.346 52.896
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 42.9519252 2.8823421 14.902 < 2e-16 ***
## perc.alumni 1.5681680 0.3905839 4.015 6.53e-05 ***
## I(perc.alumni^2) -0.0276219 0.0151030 -1.829 0.0678 .
## I(perc.alumni^3) 0.0002297 0.0001706 1.346 0.1787
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.91 on 773 degrees of freedom
## Multiple R-squared: 0.2499, Adjusted R-squared: 0.247
## F-statistic: 85.84 on 3 and 773 DF, p-value: < 2.2e-16### O termo de ordem cúbica não tem significância estatística. Vamos seguir
### a análise com o modelo quadrático.\[ \hat{Grad.Rate} = 42.9519252 + 1.5681680 \times perc.alumni -0.0276219 \times perc.alumni^2 + 0.0002297 \times perc.alumni^3 \] de acordo com os valores de p é significativo, então manteria essa variavel
### Vamos extrair alguns elementos do modelo ajustado
ajuste2$coefficients ### Estimativas dos parâmetros## (Intercept) perc.alumni I(perc.alumni^2)
## 45.896244242 1.085947860 -0.007651754ajuste2$residuals ### Resíduos ordeinários, resultado muito longo, então resumido a seguirhead(ajuste2$residuals) ## Abilene Christian University Adelphi University
## 2.174234 -5.312561
## Adrian College Agnes Scott College
## -17.588102 -16.601064
## Alaska Pacific University Albertson College
## -33.037533 -1.915809tail(ajuste2$residuals)## Worcester Polytechnic Institute Worcester State College
## 8.026956 -19.599771
## Xavier University Xavier University of Louisiana
## 10.792707 -15.554500
## Yale University York College of Pennsylvania
## 18.264171 28.696191ajuste2$fitted.values ### Valores ajustados pelo modelo, resultado muito longo, então resumido a seguirhead(ajuste2$fitted.values) ## Abilene Christian University Adelphi University
## 57.82577 61.31256
## Adrian College Agnes Scott College
## 71.58810 75.60106
## Alaska Pacific University Albertson College
## 48.03753 56.91581tail(ajuste2$fitted.values)## Worcester Polytechnic Institute Worcester State College
## 73.97304 59.59977
## Xavier University Xavier University of Louisiana
## 72.20729 64.55450
## Yale University York College of Pennsylvania
## 80.73583 70.30381model.matrix(ajuste2) ### Matriz do modelo (matriz X)## (Intercept) perc.alumni
## Abilene Christian University 1 12
## Adelphi University 1 16
## Adrian College 1 30
## Agnes Scott College 1 37
## Alaska Pacific University 1 2
## Albertson College 1 11
## Albertus Magnus College 1 26
## Albion College 1 37
## Albright College 1 23
## Alderson-Broaddus College 1 15
## Alfred University 1 31
## Allegheny College 1 41
## Allentown Coll. of St. Francis de Sales 1 21
## Alma College 1 32
## Alverno College 1 26
## American International College 1 19
## Amherst College 1 63
## Anderson University 1 14
## Andrews University 1 18
## Angelo State University 1 5
## Antioch University 1 35
## Appalachian State University 1 14
## Aquinas College 1 25
## Arizona State University Main campus 1 5
## Arkansas College (Lyon College) 1 24
## Arkansas Tech University 1 5
## Assumption College 1 30
## Auburn University-Main Campus 1 18
## Augsburg College 1 31
## Augustana College IL 1 40
## Augustana College 1 30
## Austin College 1 33
## Averett College 1 11
## Baker University 1 21
## Baldwin-Wallace College 1 20
## Barat College 1 35
## Bard College 1 30
## Barnard College 1 33
## Barry University 1 11
## Baylor University 1 38
## Beaver College 1 30
## Bellarmine College 1 31
## Belmont Abbey College 1 10
## Belmont University 1 19
## Beloit College 1 26
## Bemidji State University 1 16
## Benedictine College 1 18
## Bennington College 1 33
## Bentley College 1 20
## Berry College 1 17
## Bethany College 1 29
## Bethel College KS 1 32
## Bethel College 1 13
## Bethune Cookman College 1 9
## Birmingham-Southern College 1 34
## Blackburn College 1 53
## Bloomsburg Univ. of Pennsylvania 1 19
## Bluefield College 1 3
## Bluffton College 1 19
## Boston University 1 16
## Bowdoin College 1 52
## Bowling Green State University 1 14
## Bradford College 1 21
## Bradley University 1 21
## Brandeis University 1 24
## Brenau University 1 12
## Brewton-Parker College 1 10
## Briar Cliff College 1 26
## Bridgewater College 1 24
## Brigham Young University at Provo 1 40
## Brown University 1 39
## Bryn Mawr College 1 49
## Bucknell University 1 36
## Buena Vista College 1 10
## Butler University 1 29
## Cabrini College 1 36
## Caldwell College 1 25
## California Lutheran University 1 17
## California Polytechnic-San Luis 1 13
## California State University at Fresno 1 8
## Calvin College 1 41
## Campbell University 1 34
## Campbellsville College 1 13
## Canisius College 1 26
## Capital University 1 27
## Capitol College 1 24
## Carleton College 1 60
## Carnegie Mellon University 1 31
## Carroll College 1 25
## Carson-Newman College 1 16
## Carthage College 1 22
## Case Western Reserve University 1 29
## Castleton State College 1 8
## Catawba College 1 27
## Catholic University of America 1 18
## Cazenovia College 1 20
## Cedar Crest College 1 39
## Cedarville College 1 34
## Centenary College 1 20
## Centenary College of Louisiana 1 25
## Center for Creative Studies 1 4
## Central College 1 29
## Central Connecticut State University 1 4
## Central Missouri State University 1 4
## Central Washington University 1 0
## Central Wesleyan College 1 18
## Centre College 1 60
## Chapman University 1 6
## Chatham College 1 37
## Chestnut Hill College 1 29
## Christendom College 1 17
## Christian Brothers University 1 24
## Christopher Newport University 1 16
## Claflin College 1 31
## Claremont McKenna College 1 52
## Clark University 1 35
## Clarke College 1 27
## Clarkson University 1 32
## Clemson University 1 17
## Clinch Valley Coll. of the Univ. of Virginia 1 9
## Coe College 1 32
## Coker College 1 39
## Colby College 1 41
## Colgate University 1 45
## College Misericordia 1 23
## College of Charleston 1 18
## College of Mount St. Joseph 1 35
## College of Mount St. Vincent 1 35
## College of Notre Dame 1 7
## College of Notre Dame of Maryland 1 32
## College of Saint Benedict 1 26
## College of Saint Catherine 1 32
## College of Saint Elizabeth 1 23
## College of Saint Rose 1 28
## College of Santa Fe 1 7
## College of St. Joseph 1 19
## College of St. Scholastica 1 33
## College of the Holy Cross 1 55
## College of William and Mary 1 31
## College of Wooster 1 43
## Colorado College 1 51
## Colorado State University 1 10
## Columbia College MO 1 2
## Columbia College 1 34
## Columbia University 1 21
## Concordia College at St. Paul 1 18
## Concordia Lutheran College 1 9
## Concordia University CA 1 13
## Concordia University 1 13
## Connecticut College 1 40
## Converse College 1 31
## Cornell College 1 31
## Creighton University 1 32
## Culver-Stockton College 1 28
## Cumberland College 1 4
## D'Youville College 1 42
## Dana College 1 25
## Daniel Webster College 1 10
## Dartmouth College 1 49
## Davidson College 1 46
## Defiance College 1 19
## Delta State University 1 16
## Denison University 1 45
## DePauw University 1 31
## Dickinson College 1 39
## Dickinson State University 1 28
## Dillard University 1 12
## Doane College 1 42
## Dominican College of Blauvelt 1 5
## Dordt College 1 17
## Dowling College 1 7
## Drake University 1 24
## Drew University 1 28
## Drury College 1 35
## Duke University 1 44
## Earlham College 1 46
## East Carolina University 1 18
## East Tennessee State University 1 9
## East Texas Baptist University 1 7
## Eastern College 1 22
## Eastern Connecticut State University 1 14
## Eastern Illinois University 1 5
## Eastern Mennonite College 1 29
## Eastern Nazarene College 1 17
## Eckerd College 1 26
## Elizabethtown College 1 25
## Elmira College 1 21
## Elms College 1 21
## Elon College 1 34
## Embry Riddle Aeronautical University 1 4
## Emory & Henry College 1 51
## Emory University 1 28
## Emporia State University 1 4
## Erskine College 1 47
## Eureka College 1 31
## Evergreen State College 1 14
## Fairfield University 1 30
## Fayetteville State University 1 10
## Ferrum College 1 9
## Flagler College 1 9
## Florida Institute of Technology 1 7
## Florida International University 1 20
## Florida Southern College 1 10
## Florida State University 1 15
## Fontbonne College 1 24
## Fordham University 1 14
## Fort Lewis College 1 6
## Francis Marion University 1 8
## Franciscan University of Steubenville 1 8
## Franklin College 1 37
## Franklin Pierce College 1 16
## Freed-Hardeman University 1 13
## Fresno Pacific College 1 14
## Furman University 1 28
## Gannon University 1 18
## Gardner Webb University 1 12
## Geneva College 1 26
## George Fox College 1 22
## George Mason University 1 7
## George Washington University 1 15
## Georgetown College 1 28
## Georgetown University 1 27
## Georgia Institute of Technology 1 33
## Georgia State University 1 10
## Georgian Court College 1 27
## Gettysburg College 1 32
## Goldey Beacom College 1 4
## Gonzaga University 1 32
## Gordon College 1 32
## Goshen College 1 46
## Goucher College 1 34
## Grace College and Seminary 1 26
## Graceland College 1 24
## Grand Valley State University 1 9
## Green Mountain College 1 24
## Greensboro College 1 31
## Greenville College 1 16
## Grinnell College 1 54
## Grove City College 1 18
## Guilford College 1 30
## Gustavus Adolphus College 1 58
## Gwynedd Mercy College 1 22
## Hamilton College 1 60
## Hamline University 1 33
## Hampden - Sydney College 1 53
## Hampton University 1 9
## Hanover College 1 26
## Hardin-Simmons University 1 10
## Harding University 1 37
## Hartwick College 1 32
## Harvard University 1 52
## Harvey Mudd College 1 46
## Hastings College 1 17
## Hendrix College 1 26
## Hillsdale College 1 31
## Hiram College 1 34
## Hobart and William Smith Colleges 1 37
## Hofstra University 1 10
## Hollins College 1 48
## Hood College 1 34
## Hope College 1 40
## Houghton College 1 24
## Huntingdon College 1 9
## Huntington College 1 25
## Huron University 1 4
## Husson College 1 4
## Illinois Benedictine College 1 29
## Illinois College 1 30
## Illinois Institute of Technology 1 26
## Illinois State University 1 16
## Illinois Wesleyan University 1 34
## Immaculata College 1 33
## Incarnate Word College 1 21
## Indiana State University 1 8
## Indiana University at Bloomington 1 24
## Indiana Wesleyan University 1 15
## Iona College 1 14
## Iowa State University 1 22
## Ithaca College 1 25
## James Madison University 1 29
## Jamestown College 1 21
## Jersey City State College 1 10
## John Brown University 1 19
## John Carroll University 1 28
## Johns Hopkins University 1 38
## Johnson State College 1 15
## Judson College 1 30
## Juniata College 1 37
## Kansas State University 1 22
## Kansas Wesleyan University 1 14
## Keene State College 1 13
## Kentucky Wesleyan College 1 32
## Kenyon College 1 46
## Keuka College 1 43
## King's College 1 37
## King College 1 25
## Knox College 1 33
## La Roche College 1 14
## La Salle University 1 9
## Lafayette College 1 38
## LaGrange College 1 12
## Lake Forest College 1 19
## Lakeland College 1 25
## Lamar University 1 12
## Lambuth University 1 10
## Lander University 1 11
## Lawrence University 1 57
## Le Moyne College 1 28
## Lebanon Valley College 1 30
## Lehigh University 1 43
## Lenoir-Rhyne College 1 20
## Lesley College 1 18
## LeTourneau University 1 23
## Lewis and Clark College 1 21
## Lewis University 1 10
## Lincoln Memorial University 1 35
## Lincoln University 1 8
## Lindenwood College 1 9
## Linfield College 1 34
## Livingstone College 1 16
## Lock Haven University of Pennsylvania 1 14
## Longwood College 1 23
## Loras College 1 24
## Louisiana College 1 11
## Louisiana State University at Baton Rouge 1 11
## Louisiana Tech University 1 13
## Loyola College 1 27
## Loyola Marymount University 1 10
## Loyola University 1 14
## Loyola University Chicago 1 15
## Luther College 1 38
## Lycoming College 1 32
## Lynchburg College 1 24
## Lyndon State College 1 15
## Macalester College 1 37
## MacMurray College 1 33
## Malone College 1 16
## Manchester College 1 20
## Manhattan College 1 25
## Manhattanville College 1 24
## Mankato State University 1 11
## Marian College of Fond du Lac 1 21
## Marietta College 1 30
## Marist College 1 34
## Marquette University 1 25
## Marshall University 1 10
## Mary Baldwin College 1 50
## Mary Washington College 1 30
## Marymount College Tarrytown 1 30
## Marymount Manhattan College 1 20
## Marymount University 1 17
## Maryville College 1 43
## Maryville University 1 13
## Marywood College 1 30
## Massachusetts Institute of Technology 1 35
## Mayville State University 1 11
## McKendree College 1 21
## McMurry University 1 11
## McPherson College 1 45
## Mercer University 1 15
## Mercyhurst College 1 29
## Meredith College 1 33
## Merrimack College 1 22
## Mesa State College 1 12
## Messiah College 1 30
## Miami University at Oxford 1 20
## Michigan State University 1 9
## Michigan Technological University 1 25
## MidAmerica Nazarene College 1 20
## Millersville University of Penn. 1 20
## Milligan College 1 16
## Millikin University 1 25
## Millsaps College 1 38
## Milwaukee School of Engineering 1 23
## Mississippi College 1 18
## Mississippi State University 1 20
## Mississippi University for Women 1 8
## Missouri Southern State College 1 9
## Missouri Valley College 1 16
## Monmouth College IL 1 43
## Monmouth College 1 15
## Montana College of Mineral Sci. & Tech. 1 31
## Montana State University 1 8
## Montclair State University 1 9
## Montreat-Anderson College 1 4
## Moorhead State University 1 19
## Moravian College 1 28
## Morehouse College 1 10
## Morningside College 1 32
## Morris College 1 34
## Mount Holyoke College 1 51
## Mount Marty College 1 38
## Mount Mary College 1 26
## Mount Mercy College 1 30
## Mount Saint Clare College 1 43
## Mount Saint Mary's College 1 36
## Mount Saint Mary College 1 23
## Mount St. Mary's College 1 19
## Mount Union College 1 35
## Mount Vernon Nazarene College 1 7
## Muhlenberg College 1 39
## Murray State University 1 27
## Muskingum College 1 27
## National-Louis University 1 2
## Nazareth College of Rochester 1 24
## New Jersey Institute of Technology 1 19
## New Mexico Institute of Mining and Tech. 1 11
## New York University 1 16
## Newberry College 1 32
## Niagara University 1 20
## North Adams State College 1 17
## North Carolina A. & T. State University 1 9
## North Carolina State University at Raleigh 1 21
## North Carolina Wesleyan College 1 11
## North Central College 1 33
## North Dakota State University 1 24
## North Park College 1 24
## Northeast Missouri State University 1 13
## Northeastern University 1 17
## Northern Arizona University 1 7
## Northern Illinois University 1 11
## Northwest Missouri State University 1 23
## Northwest Nazarene College 1 20
## Northwestern College 1 34
## Northwestern University 1 25
## Norwich University 1 22
## Notre Dame College 1 26
## Oakland University 1 13
## Oberlin College 1 47
## Occidental College 1 30
## Oglethorpe University 1 27
## Ohio Northern University 1 31
## Ohio University 1 13
## Ohio Wesleyan University 1 32
## Oklahoma Baptist University 1 18
## Oklahoma Christian University 1 8
## Oklahoma State University 1 14
## Otterbein College 1 30
## Ouachita Baptist University 1 10
## Our Lady of the Lake University 1 25
## Pace University 1 10
## Pacific Lutheran University 1 23
## Pacific Union College 1 12
## Pacific University 1 22
## Pembroke State University 1 5
## Pennsylvania State Univ. Main Campus 1 19
## Pepperdine University 1 13
## Peru State College 1 24
## Pfeiffer College 1 13
## Philadelphia Coll. of Textiles and Sci. 1 15
## Phillips University 1 19
## Piedmont College 1 17
## Pikeville College 1 14
## Pitzer College 1 11
## Point Loma Nazarene College 1 19
## Point Park College 1 10
## Polytechnic University 1 14
## Prairie View A. and M. University 1 1
## Presbyterian College 1 42
## Princeton University 1 54
## Providence College 1 35
## Purdue University at West Lafayette 1 15
## Queens College 1 36
## Quincy University 1 32
## Quinnipiac College 1 33
## Radford University 1 9
## Ramapo College of New Jersey 1 8
## Randolph-Macon College 1 38
## Randolph-Macon Woman's College 1 24
## Reed College 1 37
## Regis College 1 37
## Rensselaer Polytechnic Institute 1 21
## Rhodes College 1 47
## Rider University 1 23
## Ripon College 1 49
## Rivier College 1 19
## Roanoke College 1 26
## Rockhurst College 1 21
## Rocky Mountain College 1 27
## Roger Williams University 1 8
## Rollins College 1 23
## Rosary College 1 30
## Rowan College of New Jersey 1 6
## Rutgers at New Brunswick 1 19
## Rutgers State University at Camden 1 12
## Rutgers State University at Newark 1 16
## Sacred Heart University 1 16
## Saint Ambrose University 1 16
## Saint Anselm College 1 29
## Saint Cloud State University 1 10
## Saint Francis College IN 1 4
## Saint Francis College 1 24
## Saint John's University 1 38
## Saint Joseph's College IN 1 19
## Saint Joseph's College 1 8
## Saint Joseph's University 1 13
## Saint Joseph College 1 32
## Saint Louis University 1 19
## Saint Mary's College 1 31
## Saint Mary's College of Minnesota 1 19
## Saint Mary-of-the-Woods College 1 37
## Saint Michael's College 1 34
## Saint Olaf College 1 31
## Saint Peter's College 1 22
## Saint Vincent College 1 31
## Saint Xavier University 1 15
## Salem-Teikyo University 1 9
## Salem College 1 46
## Salisbury State University 1 18
## Samford University 1 17
## San Diego State University 1 7
## Santa Clara University 1 19
## Sarah Lawrence College 1 18
## Savannah Coll. of Art and Design 1 26
## Schreiner College 1 23
## Scripps College 1 41
## Seattle Pacific University 1 20
## Seattle University 1 16
## Seton Hall University 1 15
## Seton Hill College 1 37
## Shippensburg University of Penn. 1 13
## Shorter College 1 18
## Siena College 1 42
## Siena Heights College 1 17
## Simmons College 1 33
## Simpson College 1 36
## Sioux Falls College 1 7
## Skidmore College 1 29
## Smith College 1 44
## South Dakota State University 1 29
## Southeast Missouri State University 1 8
## Southeastern Oklahoma State Univ. 1 9
## Southern California College 1 11
## Southern Illinois University at Edwardsville 1 8
## Southern Methodist University 1 17
## Southwest Baptist University 1 13
## Southwest Missouri State University 1 11
## Southwest State University 1 31
## Southwestern Adventist College 1 12
## Southwestern College 1 12
## Southwestern University 1 35
## Spalding University 1 40
## Spelman College 1 18
## Spring Arbor College 1 9
## St. Bonaventure University 1 32
## St. John's College 1 26
## St. John Fisher College 1 29
## St. Lawrence University 1 38
## St. Martin's College 1 8
## St. Mary's College of California 1 17
## St. Mary's College of Maryland 1 23
## St. Mary's University of San Antonio 1 7
## St. Norbert College 1 36
## St. Paul's College 1 8
## St. Thomas Aquinas College 1 13
## Stephens College 1 17
## Stetson University 1 24
## Stevens Institute of Technology 1 33
## Stockton College of New Jersey 1 7
## Stonehill College 1 30
## SUNY at Albany 1 16
## SUNY at Binghamton 1 15
## SUNY at Buffalo 1 15
## SUNY at Stony Brook 1 7
## SUNY College at Brockport 1 14
## SUNY College at Oswego 1 21
## SUNY College at Buffalo 1 12
## SUNY College at Cortland 1 17
## SUNY College at Fredonia 1 10
## SUNY College at Geneseo 1 25
## SUNY College at New Paltz 1 8
## SUNY College at Plattsburgh 1 16
## SUNY College at Potsdam 1 17
## SUNY College at Purchase 1 8
## Susquehanna University 1 37
## Sweet Briar College 1 48
## Syracuse University 1 13
## Tabor College 1 15
## Talladega College 1 7
## Taylor University 1 32
## Tennessee Wesleyan College 1 16
## Texas A&M Univ. at College Station 1 29
## Texas A&M University at Galveston 1 16
## Texas Christian University 1 23
## Texas Lutheran College 1 24
## Texas Southern University 1 21
## Texas Wesleyan University 1 10
## The Citadel 1 17
## Thiel College 1 16
## Tiffin University 1 40
## Transylvania University 1 41
## Trenton State College 1 6
## Tri-State University 1 24
## Trinity College CT 1 48
## Trinity College DC 1 37
## Trinity College VT 1 26
## Trinity University 1 20
## Tulane University 1 21
## Tusculum College 1 28
## Tuskegee University 1 7
## Union College KY 1 9
## Union College NY 1 49
## Univ. of Wisconsin at OshKosh 1 14
## University of Alabama at Birmingham 1 16
## University of Arkansas at Fayetteville 1 10
## University of California at Berkeley 1 10
## University of California at Irvine 1 11
## University of Central Florida 1 9
## University of Charleston 1 10
## University of Chicago 1 36
## University of Cincinnati 1 6
## University of Connecticut at Storrs 1 16
## University of Dallas 1 26
## University of Dayton 1 25
## University of Delaware 1 15
## University of Denver 1 21
## University of Detroit Mercy 1 14
## University of Dubuque 1 18
## University of Evansville 1 26
## University of Florida 1 20
## University of Georgia 1 22
## University of Hartford 1 9
## University of Hawaii at Manoa 1 6
## University of Illinois - Urbana 1 13
## University of Illinois at Chicago 1 6
## University of Indianapolis 1 23
## University of Kansas 1 17
## University of La Verne 1 23
## University of Louisville 1 24
## University of Maine at Farmington 1 26
## University of Maine at Machias 1 4
## University of Maine at Presque Isle 1 11
## University of Maryland at Baltimore County 1 6
## University of Maryland at College Park 1 12
## University of Massachusetts at Amherst 1 15
## University of Massachusetts at Dartmouth 1 20
## University of Miami 1 17
## University of Michigan at Ann Arbor 1 26
## University of Minnesota at Duluth 1 11
## University of Minnesota at Morris 1 16
## University of Minnesota Twin Cities 1 37
## University of Mississippi 1 14
## University of Missouri at Columbia 1 15
## University of Missouri at Rolla 1 23
## University of Missouri at Saint Louis 1 15
## University of Mobile 1 4
## University of Montevallo 1 8
## University of Nebraska at Lincoln 1 48
## University of New England 1 13
## University of New Hampshire 1 16
## University of North Carolina at Asheville 1 11
## University of North Carolina at Chapel Hill 1 23
## University of North Carolina at Charlotte 1 7
## University of North Carolina at Greensboro 1 17
## University of North Carolina at Wilmington 1 15
## University of North Dakota 1 16
## University of North Florida 1 14
## University of North Texas 1 6
## University of Northern Colorado 1 8
## University of Northern Iowa 1 26
## University of Notre Dame 1 46
## University of Oklahoma 1 11
## University of Oregon 1 13
## University of Pennsylvania 1 38
## University of Pittsburgh-Main Campus 1 10
## University of Portland 1 17
## University of Puget Sound 1 17
## University of Rhode Island 1 7
## University of Richmond 1 32
## University of Rochester 1 23
## University of San Diego 1 13
## University of San Francisco 1 8
## University of Sci. and Arts of Oklahoma 1 3
## University of Scranton 1 41
## University of South Carolina at Aiken 1 3
## University of South Carolina at Columbia 1 18
## University of South Florida 1 7
## University of Southern California 1 10
## University of Southern Colorado 1 0
## University of Southern Indiana 1 21
## University of Southern Mississippi 1 23
## University of St. Thomas MN 1 13
## University of St. Thomas TX 1 17
## University of Tennessee at Knoxville 1 22
## University of Texas at Arlington 1 4
## University of Texas at Austin 1 11
## University of Texas at San Antonio 1 3
## University of the Arts 1 9
## University of the Pacific 1 14
## University of the South 1 52
## University of Tulsa 1 10
## University of Utah 1 9
## University of Vermont 1 10
## University of Virginia 1 22
## University of Washington 1 10
## University of West Florida 1 12
## University of Wisconsin-Stout 1 17
## University of Wisconsin-Superior 1 15
## University of Wisconsin-Whitewater 1 16
## University of Wisconsin at Green Bay 1 1
## University of Wisconsin at Madison 1 20
## University of Wisconsin at Milwaukee 1 8
## University of Wyoming 1 13
## Upper Iowa University 1 19
## Ursinus College 1 40
## Ursuline College 1 15
## Valley City State University 1 25
## Valparaiso University 1 23
## Vanderbilt University 1 26
## Vassar College 1 39
## Villanova University 1 24
## Virginia Commonwealth University 1 11
## Virginia State University 1 11
## Virginia Tech 1 20
## Virginia Union University 1 8
## Virginia Wesleyan College 1 14
## Viterbo College 1 31
## Voorhees College 1 3
## Wabash College 1 55
## Wagner College 1 23
## Wake Forest University 1 37
## Walsh University 1 33
## Warren Wilson College 1 20
## Wartburg College 1 37
## Washington and Jefferson College 1 40
## Washington and Lee University 1 45
## Washington College 1 37
## Washington State University 1 30
## Washington University 1 31
## Wayne State College 1 29
## Waynesburg College 1 26
## Webber College 1 4
## Webster University 1 14
## Wellesley College 1 51
## Wells College 1 42
## Wentworth Institute of Technology 1 8
## Wesley College 1 15
## Wesleyan University 1 39
## West Chester University of Penn. 1 16
## West Liberty State College 1 10
## West Virginia Wesleyan College 1 42
## Western Carolina University 1 9
## Western Maryland College 1 39
## Western Michigan University 1 11
## Western New England College 1 15
## Western State College of Colorado 1 4
## Western Washington University 1 10
## Westfield State College 1 20
## Westminster College MO 1 20
## Westminster College 1 41
## Westminster College of Salt Lake City 1 34
## Westmont College 1 17
## Wheaton College IL 1 40
## Westminster College PA 1 41
## Wheeling Jesuit College 1 27
## Whitman College 1 51
## Whittier College 1 29
## Whitworth College 1 20
## Widener University 1 19
## Wilkes University 1 24
## Willamette University 1 37
## William Jewell College 1 19
## William Woods University 1 16
## Williams College 1 64
## Wilson College 1 43
## Wingate College 1 8
## Winona State University 1 18
## Winthrop University 1 26
## Wisconsin Lutheran College 1 26
## Wittenberg University 1 29
## Wofford College 1 42
## Worcester Polytechnic Institute 1 34
## Worcester State College 1 14
## Xavier University 1 31
## Xavier University of Louisiana 1 20
## Yale University 1 49
## York College of Pennsylvania 1 28
## I(perc.alumni^2)
## Abilene Christian University 144
## Adelphi University 256
## Adrian College 900
## Agnes Scott College 1369
## Alaska Pacific University 4
## Albertson College 121
## Albertus Magnus College 676
## Albion College 1369
## Albright College 529
## Alderson-Broaddus College 225
## Alfred University 961
## Allegheny College 1681
## Allentown Coll. of St. Francis de Sales 441
## Alma College 1024
## Alverno College 676
## American International College 361
## Amherst College 3969
## Anderson University 196
## Andrews University 324
## Angelo State University 25
## Antioch University 1225
## Appalachian State University 196
## Aquinas College 625
## Arizona State University Main campus 25
## Arkansas College (Lyon College) 576
## Arkansas Tech University 25
## Assumption College 900
## Auburn University-Main Campus 324
## Augsburg College 961
## Augustana College IL 1600
## Augustana College 900
## Austin College 1089
## Averett College 121
## Baker University 441
## Baldwin-Wallace College 400
## Barat College 1225
## Bard College 900
## Barnard College 1089
## Barry University 121
## Baylor University 1444
## Beaver College 900
## Bellarmine College 961
## Belmont Abbey College 100
## Belmont University 361
## Beloit College 676
## Bemidji State University 256
## Benedictine College 324
## Bennington College 1089
## Bentley College 400
## Berry College 289
## Bethany College 841
## Bethel College KS 1024
## Bethel College 169
## Bethune Cookman College 81
## Birmingham-Southern College 1156
## Blackburn College 2809
## Bloomsburg Univ. of Pennsylvania 361
## Bluefield College 9
## Bluffton College 361
## Boston University 256
## Bowdoin College 2704
## Bowling Green State University 196
## Bradford College 441
## Bradley University 441
## Brandeis University 576
## Brenau University 144
## Brewton-Parker College 100
## Briar Cliff College 676
## Bridgewater College 576
## Brigham Young University at Provo 1600
## Brown University 1521
## Bryn Mawr College 2401
## Bucknell University 1296
## Buena Vista College 100
## Butler University 841
## Cabrini College 1296
## Caldwell College 625
## California Lutheran University 289
## California Polytechnic-San Luis 169
## California State University at Fresno 64
## Calvin College 1681
## Campbell University 1156
## Campbellsville College 169
## Canisius College 676
## Capital University 729
## Capitol College 576
## Carleton College 3600
## Carnegie Mellon University 961
## Carroll College 625
## Carson-Newman College 256
## Carthage College 484
## Case Western Reserve University 841
## Castleton State College 64
## Catawba College 729
## Catholic University of America 324
## Cazenovia College 400
## Cedar Crest College 1521
## Cedarville College 1156
## Centenary College 400
## Centenary College of Louisiana 625
## Center for Creative Studies 16
## Central College 841
## Central Connecticut State University 16
## Central Missouri State University 16
## Central Washington University 0
## Central Wesleyan College 324
## Centre College 3600
## Chapman University 36
## Chatham College 1369
## Chestnut Hill College 841
## Christendom College 289
## Christian Brothers University 576
## Christopher Newport University 256
## Claflin College 961
## Claremont McKenna College 2704
## Clark University 1225
## Clarke College 729
## Clarkson University 1024
## Clemson University 289
## Clinch Valley Coll. of the Univ. of Virginia 81
## Coe College 1024
## Coker College 1521
## Colby College 1681
## Colgate University 2025
## College Misericordia 529
## College of Charleston 324
## College of Mount St. Joseph 1225
## College of Mount St. Vincent 1225
## College of Notre Dame 49
## College of Notre Dame of Maryland 1024
## College of Saint Benedict 676
## College of Saint Catherine 1024
## College of Saint Elizabeth 529
## College of Saint Rose 784
## College of Santa Fe 49
## College of St. Joseph 361
## College of St. Scholastica 1089
## College of the Holy Cross 3025
## College of William and Mary 961
## College of Wooster 1849
## Colorado College 2601
## Colorado State University 100
## Columbia College MO 4
## Columbia College 1156
## Columbia University 441
## Concordia College at St. Paul 324
## Concordia Lutheran College 81
## Concordia University CA 169
## Concordia University 169
## Connecticut College 1600
## Converse College 961
## Cornell College 961
## Creighton University 1024
## Culver-Stockton College 784
## Cumberland College 16
## D'Youville College 1764
## Dana College 625
## Daniel Webster College 100
## Dartmouth College 2401
## Davidson College 2116
## Defiance College 361
## Delta State University 256
## Denison University 2025
## DePauw University 961
## Dickinson College 1521
## Dickinson State University 784
## Dillard University 144
## Doane College 1764
## Dominican College of Blauvelt 25
## Dordt College 289
## Dowling College 49
## Drake University 576
## Drew University 784
## Drury College 1225
## Duke University 1936
## Earlham College 2116
## East Carolina University 324
## East Tennessee State University 81
## East Texas Baptist University 49
## Eastern College 484
## Eastern Connecticut State University 196
## Eastern Illinois University 25
## Eastern Mennonite College 841
## Eastern Nazarene College 289
## Eckerd College 676
## Elizabethtown College 625
## Elmira College 441
## Elms College 441
## Elon College 1156
## Embry Riddle Aeronautical University 16
## Emory & Henry College 2601
## Emory University 784
## Emporia State University 16
## Erskine College 2209
## Eureka College 961
## Evergreen State College 196
## Fairfield University 900
## Fayetteville State University 100
## Ferrum College 81
## Flagler College 81
## Florida Institute of Technology 49
## Florida International University 400
## Florida Southern College 100
## Florida State University 225
## Fontbonne College 576
## Fordham University 196
## Fort Lewis College 36
## Francis Marion University 64
## Franciscan University of Steubenville 64
## Franklin College 1369
## Franklin Pierce College 256
## Freed-Hardeman University 169
## Fresno Pacific College 196
## Furman University 784
## Gannon University 324
## Gardner Webb University 144
## Geneva College 676
## George Fox College 484
## George Mason University 49
## George Washington University 225
## Georgetown College 784
## Georgetown University 729
## Georgia Institute of Technology 1089
## Georgia State University 100
## Georgian Court College 729
## Gettysburg College 1024
## Goldey Beacom College 16
## Gonzaga University 1024
## Gordon College 1024
## Goshen College 2116
## Goucher College 1156
## Grace College and Seminary 676
## Graceland College 576
## Grand Valley State University 81
## Green Mountain College 576
## Greensboro College 961
## Greenville College 256
## Grinnell College 2916
## Grove City College 324
## Guilford College 900
## Gustavus Adolphus College 3364
## Gwynedd Mercy College 484
## Hamilton College 3600
## Hamline University 1089
## Hampden - Sydney College 2809
## Hampton University 81
## Hanover College 676
## Hardin-Simmons University 100
## Harding University 1369
## Hartwick College 1024
## Harvard University 2704
## Harvey Mudd College 2116
## Hastings College 289
## Hendrix College 676
## Hillsdale College 961
## Hiram College 1156
## Hobart and William Smith Colleges 1369
## Hofstra University 100
## Hollins College 2304
## Hood College 1156
## Hope College 1600
## Houghton College 576
## Huntingdon College 81
## Huntington College 625
## Huron University 16
## Husson College 16
## Illinois Benedictine College 841
## Illinois College 900
## Illinois Institute of Technology 676
## Illinois State University 256
## Illinois Wesleyan University 1156
## Immaculata College 1089
## Incarnate Word College 441
## Indiana State University 64
## Indiana University at Bloomington 576
## Indiana Wesleyan University 225
## Iona College 196
## Iowa State University 484
## Ithaca College 625
## James Madison University 841
## Jamestown College 441
## Jersey City State College 100
## John Brown University 361
## John Carroll University 784
## Johns Hopkins University 1444
## Johnson State College 225
## Judson College 900
## Juniata College 1369
## Kansas State University 484
## Kansas Wesleyan University 196
## Keene State College 169
## Kentucky Wesleyan College 1024
## Kenyon College 2116
## Keuka College 1849
## King's College 1369
## King College 625
## Knox College 1089
## La Roche College 196
## La Salle University 81
## Lafayette College 1444
## LaGrange College 144
## Lake Forest College 361
## Lakeland College 625
## Lamar University 144
## Lambuth University 100
## Lander University 121
## Lawrence University 3249
## Le Moyne College 784
## Lebanon Valley College 900
## Lehigh University 1849
## Lenoir-Rhyne College 400
## Lesley College 324
## LeTourneau University 529
## Lewis and Clark College 441
## Lewis University 100
## Lincoln Memorial University 1225
## Lincoln University 64
## Lindenwood College 81
## Linfield College 1156
## Livingstone College 256
## Lock Haven University of Pennsylvania 196
## Longwood College 529
## Loras College 576
## Louisiana College 121
## Louisiana State University at Baton Rouge 121
## Louisiana Tech University 169
## Loyola College 729
## Loyola Marymount University 100
## Loyola University 196
## Loyola University Chicago 225
## Luther College 1444
## Lycoming College 1024
## Lynchburg College 576
## Lyndon State College 225
## Macalester College 1369
## MacMurray College 1089
## Malone College 256
## Manchester College 400
## Manhattan College 625
## Manhattanville College 576
## Mankato State University 121
## Marian College of Fond du Lac 441
## Marietta College 900
## Marist College 1156
## Marquette University 625
## Marshall University 100
## Mary Baldwin College 2500
## Mary Washington College 900
## Marymount College Tarrytown 900
## Marymount Manhattan College 400
## Marymount University 289
## Maryville College 1849
## Maryville University 169
## Marywood College 900
## Massachusetts Institute of Technology 1225
## Mayville State University 121
## McKendree College 441
## McMurry University 121
## McPherson College 2025
## Mercer University 225
## Mercyhurst College 841
## Meredith College 1089
## Merrimack College 484
## Mesa State College 144
## Messiah College 900
## Miami University at Oxford 400
## Michigan State University 81
## Michigan Technological University 625
## MidAmerica Nazarene College 400
## Millersville University of Penn. 400
## Milligan College 256
## Millikin University 625
## Millsaps College 1444
## Milwaukee School of Engineering 529
## Mississippi College 324
## Mississippi State University 400
## Mississippi University for Women 64
## Missouri Southern State College 81
## Missouri Valley College 256
## Monmouth College IL 1849
## Monmouth College 225
## Montana College of Mineral Sci. & Tech. 961
## Montana State University 64
## Montclair State University 81
## Montreat-Anderson College 16
## Moorhead State University 361
## Moravian College 784
## Morehouse College 100
## Morningside College 1024
## Morris College 1156
## Mount Holyoke College 2601
## Mount Marty College 1444
## Mount Mary College 676
## Mount Mercy College 900
## Mount Saint Clare College 1849
## Mount Saint Mary's College 1296
## Mount Saint Mary College 529
## Mount St. Mary's College 361
## Mount Union College 1225
## Mount Vernon Nazarene College 49
## Muhlenberg College 1521
## Murray State University 729
## Muskingum College 729
## National-Louis University 4
## Nazareth College of Rochester 576
## New Jersey Institute of Technology 361
## New Mexico Institute of Mining and Tech. 121
## New York University 256
## Newberry College 1024
## Niagara University 400
## North Adams State College 289
## North Carolina A. & T. State University 81
## North Carolina State University at Raleigh 441
## North Carolina Wesleyan College 121
## North Central College 1089
## North Dakota State University 576
## North Park College 576
## Northeast Missouri State University 169
## Northeastern University 289
## Northern Arizona University 49
## Northern Illinois University 121
## Northwest Missouri State University 529
## Northwest Nazarene College 400
## Northwestern College 1156
## Northwestern University 625
## Norwich University 484
## Notre Dame College 676
## Oakland University 169
## Oberlin College 2209
## Occidental College 900
## Oglethorpe University 729
## Ohio Northern University 961
## Ohio University 169
## Ohio Wesleyan University 1024
## Oklahoma Baptist University 324
## Oklahoma Christian University 64
## Oklahoma State University 196
## Otterbein College 900
## Ouachita Baptist University 100
## Our Lady of the Lake University 625
## Pace University 100
## Pacific Lutheran University 529
## Pacific Union College 144
## Pacific University 484
## Pembroke State University 25
## Pennsylvania State Univ. Main Campus 361
## Pepperdine University 169
## Peru State College 576
## Pfeiffer College 169
## Philadelphia Coll. of Textiles and Sci. 225
## Phillips University 361
## Piedmont College 289
## Pikeville College 196
## Pitzer College 121
## Point Loma Nazarene College 361
## Point Park College 100
## Polytechnic University 196
## Prairie View A. and M. University 1
## Presbyterian College 1764
## Princeton University 2916
## Providence College 1225
## Purdue University at West Lafayette 225
## Queens College 1296
## Quincy University 1024
## Quinnipiac College 1089
## Radford University 81
## Ramapo College of New Jersey 64
## Randolph-Macon College 1444
## Randolph-Macon Woman's College 576
## Reed College 1369
## Regis College 1369
## Rensselaer Polytechnic Institute 441
## Rhodes College 2209
## Rider University 529
## Ripon College 2401
## Rivier College 361
## Roanoke College 676
## Rockhurst College 441
## Rocky Mountain College 729
## Roger Williams University 64
## Rollins College 529
## Rosary College 900
## Rowan College of New Jersey 36
## Rutgers at New Brunswick 361
## Rutgers State University at Camden 144
## Rutgers State University at Newark 256
## Sacred Heart University 256
## Saint Ambrose University 256
## Saint Anselm College 841
## Saint Cloud State University 100
## Saint Francis College IN 16
## Saint Francis College 576
## Saint John's University 1444
## Saint Joseph's College IN 361
## Saint Joseph's College 64
## Saint Joseph's University 169
## Saint Joseph College 1024
## Saint Louis University 361
## Saint Mary's College 961
## Saint Mary's College of Minnesota 361
## Saint Mary-of-the-Woods College 1369
## Saint Michael's College 1156
## Saint Olaf College 961
## Saint Peter's College 484
## Saint Vincent College 961
## Saint Xavier University 225
## Salem-Teikyo University 81
## Salem College 2116
## Salisbury State University 324
## Samford University 289
## San Diego State University 49
## Santa Clara University 361
## Sarah Lawrence College 324
## Savannah Coll. of Art and Design 676
## Schreiner College 529
## Scripps College 1681
## Seattle Pacific University 400
## Seattle University 256
## Seton Hall University 225
## Seton Hill College 1369
## Shippensburg University of Penn. 169
## Shorter College 324
## Siena College 1764
## Siena Heights College 289
## Simmons College 1089
## Simpson College 1296
## Sioux Falls College 49
## Skidmore College 841
## Smith College 1936
## South Dakota State University 841
## Southeast Missouri State University 64
## Southeastern Oklahoma State Univ. 81
## Southern California College 121
## Southern Illinois University at Edwardsville 64
## Southern Methodist University 289
## Southwest Baptist University 169
## Southwest Missouri State University 121
## Southwest State University 961
## Southwestern Adventist College 144
## Southwestern College 144
## Southwestern University 1225
## Spalding University 1600
## Spelman College 324
## Spring Arbor College 81
## St. Bonaventure University 1024
## St. John's College 676
## St. John Fisher College 841
## St. Lawrence University 1444
## St. Martin's College 64
## St. Mary's College of California 289
## St. Mary's College of Maryland 529
## St. Mary's University of San Antonio 49
## St. Norbert College 1296
## St. Paul's College 64
## St. Thomas Aquinas College 169
## Stephens College 289
## Stetson University 576
## Stevens Institute of Technology 1089
## Stockton College of New Jersey 49
## Stonehill College 900
## SUNY at Albany 256
## SUNY at Binghamton 225
## SUNY at Buffalo 225
## SUNY at Stony Brook 49
## SUNY College at Brockport 196
## SUNY College at Oswego 441
## SUNY College at Buffalo 144
## SUNY College at Cortland 289
## SUNY College at Fredonia 100
## SUNY College at Geneseo 625
## SUNY College at New Paltz 64
## SUNY College at Plattsburgh 256
## SUNY College at Potsdam 289
## SUNY College at Purchase 64
## Susquehanna University 1369
## Sweet Briar College 2304
## Syracuse University 169
## Tabor College 225
## Talladega College 49
## Taylor University 1024
## Tennessee Wesleyan College 256
## Texas A&M Univ. at College Station 841
## Texas A&M University at Galveston 256
## Texas Christian University 529
## Texas Lutheran College 576
## Texas Southern University 441
## Texas Wesleyan University 100
## The Citadel 289
## Thiel College 256
## Tiffin University 1600
## Transylvania University 1681
## Trenton State College 36
## Tri-State University 576
## Trinity College CT 2304
## Trinity College DC 1369
## Trinity College VT 676
## Trinity University 400
## Tulane University 441
## Tusculum College 784
## Tuskegee University 49
## Union College KY 81
## Union College NY 2401
## Univ. of Wisconsin at OshKosh 196
## University of Alabama at Birmingham 256
## University of Arkansas at Fayetteville 100
## University of California at Berkeley 100
## University of California at Irvine 121
## University of Central Florida 81
## University of Charleston 100
## University of Chicago 1296
## University of Cincinnati 36
## University of Connecticut at Storrs 256
## University of Dallas 676
## University of Dayton 625
## University of Delaware 225
## University of Denver 441
## University of Detroit Mercy 196
## University of Dubuque 324
## University of Evansville 676
## University of Florida 400
## University of Georgia 484
## University of Hartford 81
## University of Hawaii at Manoa 36
## University of Illinois - Urbana 169
## University of Illinois at Chicago 36
## University of Indianapolis 529
## University of Kansas 289
## University of La Verne 529
## University of Louisville 576
## University of Maine at Farmington 676
## University of Maine at Machias 16
## University of Maine at Presque Isle 121
## University of Maryland at Baltimore County 36
## University of Maryland at College Park 144
## University of Massachusetts at Amherst 225
## University of Massachusetts at Dartmouth 400
## University of Miami 289
## University of Michigan at Ann Arbor 676
## University of Minnesota at Duluth 121
## University of Minnesota at Morris 256
## University of Minnesota Twin Cities 1369
## University of Mississippi 196
## University of Missouri at Columbia 225
## University of Missouri at Rolla 529
## University of Missouri at Saint Louis 225
## University of Mobile 16
## University of Montevallo 64
## University of Nebraska at Lincoln 2304
## University of New England 169
## University of New Hampshire 256
## University of North Carolina at Asheville 121
## University of North Carolina at Chapel Hill 529
## University of North Carolina at Charlotte 49
## University of North Carolina at Greensboro 289
## University of North Carolina at Wilmington 225
## University of North Dakota 256
## University of North Florida 196
## University of North Texas 36
## University of Northern Colorado 64
## University of Northern Iowa 676
## University of Notre Dame 2116
## University of Oklahoma 121
## University of Oregon 169
## University of Pennsylvania 1444
## University of Pittsburgh-Main Campus 100
## University of Portland 289
## University of Puget Sound 289
## University of Rhode Island 49
## University of Richmond 1024
## University of Rochester 529
## University of San Diego 169
## University of San Francisco 64
## University of Sci. and Arts of Oklahoma 9
## University of Scranton 1681
## University of South Carolina at Aiken 9
## University of South Carolina at Columbia 324
## University of South Florida 49
## University of Southern California 100
## University of Southern Colorado 0
## University of Southern Indiana 441
## University of Southern Mississippi 529
## University of St. Thomas MN 169
## University of St. Thomas TX 289
## University of Tennessee at Knoxville 484
## University of Texas at Arlington 16
## University of Texas at Austin 121
## University of Texas at San Antonio 9
## University of the Arts 81
## University of the Pacific 196
## University of the South 2704
## University of Tulsa 100
## University of Utah 81
## University of Vermont 100
## University of Virginia 484
## University of Washington 100
## University of West Florida 144
## University of Wisconsin-Stout 289
## University of Wisconsin-Superior 225
## University of Wisconsin-Whitewater 256
## University of Wisconsin at Green Bay 1
## University of Wisconsin at Madison 400
## University of Wisconsin at Milwaukee 64
## University of Wyoming 169
## Upper Iowa University 361
## Ursinus College 1600
## Ursuline College 225
## Valley City State University 625
## Valparaiso University 529
## Vanderbilt University 676
## Vassar College 1521
## Villanova University 576
## Virginia Commonwealth University 121
## Virginia State University 121
## Virginia Tech 400
## Virginia Union University 64
## Virginia Wesleyan College 196
## Viterbo College 961
## Voorhees College 9
## Wabash College 3025
## Wagner College 529
## Wake Forest University 1369
## Walsh University 1089
## Warren Wilson College 400
## Wartburg College 1369
## Washington and Jefferson College 1600
## Washington and Lee University 2025
## Washington College 1369
## Washington State University 900
## Washington University 961
## Wayne State College 841
## Waynesburg College 676
## Webber College 16
## Webster University 196
## Wellesley College 2601
## Wells College 1764
## Wentworth Institute of Technology 64
## Wesley College 225
## Wesleyan University 1521
## West Chester University of Penn. 256
## West Liberty State College 100
## West Virginia Wesleyan College 1764
## Western Carolina University 81
## Western Maryland College 1521
## Western Michigan University 121
## Western New England College 225
## Western State College of Colorado 16
## Western Washington University 100
## Westfield State College 400
## Westminster College MO 400
## Westminster College 1681
## Westminster College of Salt Lake City 1156
## Westmont College 289
## Wheaton College IL 1600
## Westminster College PA 1681
## Wheeling Jesuit College 729
## Whitman College 2601
## Whittier College 841
## Whitworth College 400
## Widener University 361
## Wilkes University 576
## Willamette University 1369
## William Jewell College 361
## William Woods University 256
## Williams College 4096
## Wilson College 1849
## Wingate College 64
## Winona State University 324
## Winthrop University 676
## Wisconsin Lutheran College 676
## Wittenberg University 841
## Wofford College 1764
## Worcester Polytechnic Institute 1156
## Worcester State College 196
## Xavier University 961
## Xavier University of Louisiana 400
## Yale University 2401
## York College of Pennsylvania 784
## attr(,"assign")
## [1] 0 1 2vcov(ajuste2) ### Matriz de variâncias e covariâncias dos estimadores.## (Intercept) perc.alumni I(perc.alumni^2)
## (Intercept) 3.526090777 -0.2679031981 4.254628e-03
## perc.alumni -0.267903198 0.0242157266 -4.217689e-04
## I(perc.alumni^2) 0.004254628 -0.0004217689 7.959571e-06### Vamos visualizar o ajuste do modelo
ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
stat_smooth(method = "lm", formula = y ~ x + I(x^2)) +
theme_bw(base_size = 14)
### Extraindo os intervalos de confiança (95%) para os parâmetros
confint(ajuste2)## 2.5 % 97.5 %
## (Intercept) 42.2100816 49.582406847
## perc.alumni 0.7804723 1.391423440
## I(perc.alumni^2) -0.0131900 -0.002113503### vamos realizar algumas predições. Considere faculdades com os seguintes
### percentuais de ex-alunos contribuintes: 13, 28 e 45
new_data <- data.frame(perc.alumni = c(13,28,45))
predict(ajuste2, newdata = new_data, se.fit = TRUE)## $fit
## 1 2 3
## 58.72042 70.30381 79.26910
##
## $se.fit
## 1 2 3
## 0.6820636 0.7449543 1.2061247
##
## $df
## [1] 774
##
## $residual.scale
## [1] 14.91414### Estimativas pontuais e erros padrões (nota: erros padrões para a resposta
### média)
predict(ajuste2, newdata = new_data, interval = 'confidence') # para grupo com essa média## fit lwr upr
## 1 58.72042 57.38151 60.05933
## 2 70.30381 68.84144 71.76618
## 3 79.26910 76.90143 81.63676### fit = estimando os formados a partir de um número de doadores (13,28,45). lwr e upr = limites do 95% de confiaça.
### Estimativas pontiais e intervalos de confiança (95%) para a resposta **média**.
predict(ajuste2, newdata = new_data, interval = 'prediction') ## Para caso isolado## fit lwr upr
## 1 58.72042 29.41286 88.02798
## 2 70.30381 40.99035 99.61727
## 3 79.26910 49.89656 108.64164### fit = estimando os formados a partir de um número de doadores (13,28,45). lwr e upr = limites do 95% de confiaça
### Estimativas pontiais e intervalos de confiança (95%) para a predição de **uma nova observação**.### Vamos plotar as bandas de confiança e predição. Para isso, vamos preparar
### uma base de dados com os valores ajustados e os ICs(95%) para a resposta
### média e para as predições.
pred_int <- predict(ajuste2, interval="prediction")
med_int <- predict(ajuste2, interval="confidence")
data_pred <- data.frame(pred_lwr = pred_int[,'lwr'], pred_upr = pred_int[,'upr'],
med_lwr = med_int[,'lwr'], med_upr = med_int[,'upr'],
fit = med_int[,'fit'], perc.alumni = College$perc.alumni,
Grad.Rate = College$Grad.Rate)
ggplot(data_pred, aes(x = perc.alumni, y = Grad.Rate))+
geom_point() +
geom_line(aes(y=med_lwr), color = "red", linetype = "dashed", linewidth = 1.25) +
geom_line(aes(y=med_upr), color = "red", linetype = "dashed", linewidth = 1.25) +
geom_line(aes(y=pred_lwr), color = "green", linetype = "dashed", linewidth = 1.25) +
geom_line(aes(y=pred_upr), color = "green", linetype = "dashed", linewidth = 1.25) +
geom_line(aes(y=fit), color = "black", linetype = "dashed", linewidth = 1.25) +
theme_bw(base_size = 14)### Diagnóstico do ajuste (análise de resíduos)
### Vamos produzir alguns gráficos para os resíduos
fit_aj2 <- fitted(ajuste2) ### Vetor de valores ajustados
resid_aj2 <- rstandard(ajuste2) ### Vetor de resíduos padronizados
data_fit <- data.frame(y = College$Grad.Rate, fit_aj2, resid_aj2)
ggplot(data_fit, aes(x=y, y=fit_aj2)) + geom_point() + stat_smooth(method="lm") +
theme_bw(base_size = 14)
### Gráfico de valores observados versus valores ajustados.
ggplot(ajuste2, aes(x=fit_aj2, y=resid_aj2)) + geom_point() +
stat_smooth(method="loess") + geom_hline(yintercept=0, col="red", linetype="dashed") +
theme_bw(base_size = 14) +
xlab('Valores ajustados') +
ylab('Resíduos')
### Gráfico de resíduos versus valores ajustados
qqPlot(ajuste2)
### Gráfico quantil-quantil para os resíduos### Parte 2 - Regressão linear múltipla, incluindo todas as variáveis da base como explicativas (exceto a taxa de formação, que é a resposta)
ajuste_p2 <- lm(Grad.Rate ~., data = College)
### Ajuste da regressão linear múltipla. A especificação "~." indica que
### todas as demais variáveis da base devem ser incluídas como explicativas.
### A título de ilustração, se quiséssemos ajustar um modelo apenas com as
### variáveis "Private", "Apps" e "Accept":
summary(ajuste_p2) ##
## Call:
## lm(formula = Grad.Rate ~ ., data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -53.897 -7.132 -0.292 7.213 54.056
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.8736716 4.8480858 6.987 6.15e-12 ***
## PrivateYes 3.3813758 1.6965147 1.993 0.046605 *
## Apps 0.0012984 0.0004418 2.939 0.003390 **
## Accept -0.0006961 0.0008627 -0.807 0.419995
## Enroll 0.0021593 0.0023081 0.936 0.349814
## Top10perc 0.0548964 0.0717587 0.765 0.444501
## Top25perc 0.1351288 0.0549667 2.458 0.014179 *
## F.Undergrad -0.0004712 0.0004008 -1.176 0.240138
## P.Undergrad -0.0014836 0.0003902 -3.802 0.000155 ***
## Outstate 0.0010174 0.0002334 4.359 1.49e-05 ***
## Room.Board 0.0019143 0.0005908 3.240 0.001246 **
## Books -0.0022205 0.0029168 -0.761 0.446739
## Personal -0.0016635 0.0007698 -2.161 0.031000 *
## PhD 0.0872827 0.0568102 1.536 0.124859
## Terminal -0.0747023 0.0623172 -1.199 0.231002
## S.F.Ratio 0.0758222 0.1593102 0.476 0.634254
## perc.alumni 0.2793343 0.0491750 5.680 1.91e-08 ***
## Expend -0.0004565 0.0001542 -2.961 0.003163 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.75 on 759 degrees of freedom
## Multiple R-squared: 0.4615, Adjusted R-squared: 0.4495
## F-statistic: 38.27 on 17 and 759 DF, p-value: < 2.2e-16ajuste_p2 <- lm(Grad.Rate ~.-Top25perc, data = College)
## Todas menos a top 25. pq o top 10 e top 25 "se sobrepoe"
## "ajustado os efeitos das outras variaveis"
summary(ajuste_p2) ##
## Call:
## lm(formula = Grad.Rate ~ . - Top25perc, data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.786 -7.120 -0.257 7.171 54.050
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 36.3495104 4.7580380 7.640 6.57e-14 ***
## PrivateYes 3.3824039 1.7021347 1.987 0.047264 *
## Apps 0.0011831 0.0004407 2.685 0.007421 **
## Accept -0.0003936 0.0008568 -0.459 0.646059
## Enroll 0.0015802 0.0023036 0.686 0.492938
## Top10perc 0.1951766 0.0436553 4.471 8.98e-06 ***
## F.Undergrad -0.0003934 0.0004009 -0.981 0.326837
## P.Undergrad -0.0014830 0.0003915 -3.788 0.000164 ***
## Outstate 0.0010161 0.0002342 4.339 1.63e-05 ***
## Room.Board 0.0019288 0.0005927 3.254 0.001188 **
## Books -0.0020595 0.0029258 -0.704 0.481692
## Personal -0.0016865 0.0007723 -2.184 0.029273 *
## PhD 0.0876416 0.0569982 1.538 0.124558
## Terminal -0.0565580 0.0620836 -0.911 0.362585
## S.F.Ratio 0.0725205 0.1598322 0.454 0.650153
## perc.alumni 0.2896423 0.0491582 5.892 5.73e-09 ***
## Expend -0.0005259 0.0001521 -3.459 0.000573 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.79 on 760 degrees of freedom
## Multiple R-squared: 0.4573, Adjusted R-squared: 0.4458
## F-statistic: 40.02 on 16 and 760 DF, p-value: < 2.2e-16Exemplo de numero de comodos e area em preço do imovel. Em sepada os dois são significantes. No modelo tudo junto ajutado a area é positiva e os comodos são negativos (mais comodos numa mesma area, ou seja comodos menores, menos preço).
ajuste_p2 <- lm(Grad.Rate ~ Expend, data = College)
## Todas menos a top 25. pq o top 10 e top 25 "se sobrepoe"
## "ajustado os efeitos das outras variaveis"
summary(ajuste_p2) ##
## Call:
## lm(formula = Grad.Rate ~ Expend, data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -60.179 -10.275 0.238 10.377 55.058
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.306e+01 1.194e+00 44.42 <2e-16 ***
## Expend 1.284e-03 1.088e-04 11.80 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.83 on 775 degrees of freedom
## Multiple R-squared: 0.1524, Adjusted R-squared: 0.1513
## F-statistic: 139.3 on 1 and 775 DF, p-value: < 2.2e-16ajuste_p2_ilustrativo <- lm(Grad.Rate ~ Private + Apps + Accept, data = College)
summary(ajuste_p2_ilustrativo) ##
## Call:
## lm(formula = Grad.Rate ~ Private + Apps + Accept, data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -51.513 -8.856 0.228 10.516 50.053
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 48.5068491 1.4308933 33.900 < 2e-16 ***
## PrivateYes 17.7697914 1.3831646 12.847 < 2e-16 ***
## Apps 0.0030589 0.0004228 7.235 1.12e-12 ***
## Accept -0.0025493 0.0006842 -3.726 0.000209 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.09 on 773 degrees of freedom
## Multiple R-squared: 0.2316, Adjusted R-squared: 0.2287
## F-statistic: 77.68 on 3 and 773 DF, p-value: < 2.2e-16### Resumo do ajuste. Observe as variáveis com efeito significativo e os respectivos sinais.
### Variáveis com p-valor (Pr(>|t|)) < 0.05 podem ser consideradas com efeito
### significativo na taxa de formados. Desta forma, as variáveis com efeito
### significativo na taxa de formados são Private (Yes), Apps, Top25perc, Outstate,
### Room.Board, 0.2793343; Já as variáveis com efeito negativo na taxa de formados
### são P.Undergrad, Personal e Expend.### Apenas para fins de discussão, vamos ajustar um modelo de regressão tendo como
### única variável explicativa o gasto por aluno.
ajuste_temp <- lm(Grad.Rate ~ Expend, data = College)
summary(ajuste_temp)
### Compare o efeito do gasto por aluno na taxa de formação produzida pelo
### modelo em que ajustamos também os efeitos das demais variáveis com o
### efeito produzido pelo modelo em que as demais variáveis não são consideradas.
### Qual a diferença? Como você a justifica?par(mfrow = c(2,2))
plot(ajuste_p2, which = 1:4)
### Gráficos de resíduos.
### O gráfico do canto superior esquerdo indica que a variância dos resíduos varia
### um pouco com a média (fitted values), e que os resíduos apresentam algum desvio
### da normalidade (conforme o qqplot). Ainda, conforme o gráfico do canto inferior
### direito, algumas observações podem ser identificadas como possivelmente influentes,
### produzindo maiores valores para a distância de Cook.
### Embora tenhamos alguns indicativos (ainda que não tão severos) de falta de ajuste
### da regressão linear, para fins didáticos vamos seguir a análise com esse tipo
### de modelagem.### Agora, vamos para a etapa de seleção de covariáveis. Para isso, vamos
### usar os recursos do pacote leaps. Vamos consultar a documentação da função
### regsubsets.
help("regsubsets")
all_reg <- regsubsets(Grad.Rate ~ ., method = "exhaustive", nvmax = 18, data = College)
### Explorando TODAS as regressões possíveis
all_regplot(all_reg, scale="r2") ### Resultados baseados no R2
plot(all_reg, scale="adjr2") ### Resultados baseados no R2 ajustado
plot(all_reg, scale="bic") ### Resultados baseados no BIC
### Nesses gráficos avaliamos os resultados dos critérios para os melhores
### modelos ajustados com cada número de covariáveis.
s1 <- summary(all_reg, matrix.logical=TRUE)
s1 ### A matriz lógica permite identificar as covariáveis selecionadas
### em cada modelo.
s1$rsq ### Valores de R2 para cada um dos modelos selecionados
s1$adjr2 ### Valores de R2 ajustado para cada um dos modelos selecionados
s1$bic ### Valores de BIC para cada um dos modelos selecionados
which.max(s1$adjr2)
coef(all_reg, id = 12)
### O modelo com doze covariáveis produziu maior valor de R2 ajustado.
which.min(s1$bic)
coef(all_reg, id = 7)
### O modelo com sete covariáveis produziu menor valor de BIC.
which.max(s1$rsq)
coef(all_reg, id = 17)
### O modelo com 17 covariáveis produziu menor valor de R2 (obviamente).
### Vamos produzir alguns gráficos usando os resultados desta análise.
n_cov <- 1:17
plot(n_cov, s1$bic, type = 'b', xlab = 'Número de covariáveis',
ylab = 'BIC', las = 1, pch = 20)
axis(1,1:17)
plot(n_cov, s1$adjr2, type = 'b', xlab = 'Número de covariáveis',
ylab = 'Adjusted R2', las = 1, pch = 20)
axis(1,1:17)
plot(n_cov, s1$rsq, type = 'b', xlab = 'Número de covariáveis',
ylab = 'R2', las = 1, pch = 20)
axis(1,1:17)
### Para finalizar, vamos aplicar os algoritmos de seleção do tipo
### stepwise. Primeiramente fixando k = 2, estamos definindo o AIC como critério de
### seleção, temos:
### Método backward
step_back_AIC <- step(aj_full, direction = "backward", data = College, k = 2)
summary(step_back_AIC)
### Método forward. Para o método forward devemos definir o escopo da seleção
### (menor e maior modelo). O menor seria o modelo nulo (apenas com o intercepto),
### enquanto o maior seria o modelo com todas as covariáveis.
aj_lower <- lm(Grad.Rate~1, data = College)
aj_upper <- lm(Grad.Rate~., data = College)
formula(aj_upper)
step_for_AIC <- step(aj_lower, direction = "forward", scope=formula(aj_upper),
data = College, k = 2)
summary(step_for_AIC)
### Finalmente, o algoritmo que considera tanto exclusão quanto inclusão de
### covariáveis a cada passo
step_both_AIC <- step(aj_full, direction = "both", data = College, k = 2)
summary(step_both_AIC)
### Vamos comparar os ajustes.
data.frame(compareCoefs(step_back_AIC, step_for_AIC, step_both_AIC))
### Agora fixando k = log(n) (critério BIC):
### Método backward
step_back_BIC <- step(aj_full, direction = "backward", data = College, k = log(nrow(College)))
summary(step_back_BIC)
### Método forward. Para o método forward devemos definir o escopo da seleção
### (menor e maior modelo). O menor seria o modelo nulo (apenas com o intercepto),
### enquanto o maior seria o modelo com todas as covariáveis.
aj_lower <- lm(Grad.Rate~1, data = College)
aj_upper <- lm(Grad.Rate~., data = College)
formula(aj_upper)
step_for_BIC <- step(aj_lower, direction = "forward", scope=formula(aj_upper),
data = College, k = log(nrow(College)))
summary(step_for_BIC)
### Finalmente, o algoritmo que considera tanto exclusão quanto inclusão de
### covariáveis a cada passo
step_both_BIC <- step(aj_full, direction = "both", data = College, k = log(nrow(College)))
summary(step_both_BIC)
### Vamos comparar os ajustes.
data.frame(compareCoefs(step_back_BIC, step_for_BIC, step_both_BIC))